bounds on the capacity of ofdm under spread ... - IEEE Xplore

BOUNDS ON THE CAPACITY OF OFDM UNDER SPREAD FREQUENCY SELECTIVEFADING CHANNELSItsik BergelSchool ofEngineeringBar-Ilan Universitye-mail:bergeli@eng.biu.ac.ilSergio BenedettoDipartimento di ElettronicaPolitecnico di Torinoe-mail:benedetto@polito.itABSTRACTIn this paper we derive novel <strong>bounds</strong> on the capacity ofOFDM under spread frequency selective fading channelswithout prior knowledge of the channel (non-coherent channel).The derived <strong>bounds</strong> are characterized by a newly definedparameter termed effective coherence time, which characterizethe number of effective number of samples that can beused for channel estimation for a given SNR. We also showthat the effective coherence time is a non-increasing functionof the SNR. The presented <strong>bounds</strong> are tight for any SNR aslong as the effective coherence time is large enough.1. INTRODUCTIONThe analysis of communication systems is often performedunder the assumption ofperfect channel knowledge. In practicalsystems, however, the channel needs to be estimated fromthe communication signal itself, or from a properly designedand broadcasted pilot signal, so that this assumption is oftenunrealistic.The capacity of communication systems without priorknowledge of the channel has been investigated recently byseveral researchers. The case of flat fading channels withindependent identically distributed (iid) fading has receivedmuch attention, and although there is no closed form expression,its capacity is well characterized (e.g. [1, 2]). The frequencyflat correlated fading case was mainly analyzed as aniid fading model with channel state information (CSI), underthe assumption that the CSI was obtained from the previouslyreceived signals. The analysis of such systems revealed adifferent behavior for the low and high signal-to-noise ratio(SNR) regimes.For the low SNR regime, it was shown [3,4] that the channelcapacity is identical to the capacity when the channel isknown. I.e., the lack of channel knowledge does not reducethe capacity. But, due to the imperfect channel knowledge,the system is required to use a different signaling scheme.While in the known channel case, iid Gaussian signaling isoptimal, in the complete absence of channel knowledge theoptimal signaling must have a low duty-cycle (i.e., the proba-bility oftransmitting a zero must be large). In case ofa channelwith imperfect CSI, it was shown [4] that a system usingGaussian signaling achieves significantly less than the channelcapacity.For the high SNR regime the situation is different. In thiscase the capacity in the absence ofchannel knowledge is significantlyinferior to the capacity with perfect channel knowledge,and grows only double logarithmically with SNR [5, 6].In this case, again, the Gaussian signaling is far from optimal.In fact, it was shown [4] that for any fixed channel estimationerror variance, the mutual information ofa system usingGaussian signaling is bounded as the SNR goes to infinity,while even in the case of no-channel knowledge the channelcapacity is unbounded.While avoiding the use ofGaussian signaling may not representa problem, the use oflow duty cycle signals (that wereshown to be optimal for the iid fading case [3]) is problematicin many practical systems. In fact, as the system SNRapproaches zero, the duty cycle required to achieve capacityapproaches zero even faster (i.e., transmission of infinitepower for very short time). In order to account for the impossibilityoftransmitting such strong impulses, the analysismust include another constraint that limits the peak power ofthe signal.The most common constraint used for this purpose is aconstraint on the fourth moment of the transmitted signal(termed quadratic constraint). This constraint was used toanalyze the capacity ofthe frequency selective fading channelin the wideband limit. It was shown [7, 8] that the capacityof this channel with the quadratic constraint is a decreasingfunction of the number of resolvable paths. If the numberof resolvable paths also grows to infinity as the bandwidthgrows to infinity, then the capacity converges to zero.In this case the channel coherence time becomes the mostimportant characteristic ofthe channel, as it describes the rateof changes in the channel (and therefore characterize the capabilityto learn the channel). But, the common definition ofchannel coherence time (e.g., [9]) cannot be directly related toany capacity result. Instead, [7] had considered a block fadingmodel of length T c , while [8] had considered a more general1-4244-2482-5/08/$20.00 ©2008 IEEE 755 IEEEI2008

model and gave a novel definition for the channel coherencetime that is directly connected to the channel capacity.But, this definition ofchannel coherence time ([8]) is onlyeffective for infinite bandwidth (or equivalently at the lowSNR limit). In this paper we extend this definition, and showthat the novel definition is applicable for finite bandwidth andanySNR.In this paper we represent the finite bandwidth underspreadchannel by an OFOM under-spread channel andpresent novel <strong>bounds</strong> on the channel capacity. This <strong>bounds</strong>are characterized by a newly defined parameter termed effectivecoherence time. The novel lower and upper <strong>bounds</strong>derived in this paper are tight enough to characterize thechannel capacity as long as the effective coherence time islarge enough.2.1. Channel model2. SYSTEM MODELWe use the OFOM model [10], in which the Fourier transformofthe received signal is given by:- 6. t - 6. twhere X k = F X k and H k = F H k are the Fourier transformsofthe input vector and channel impulse response at thek-th symbol respectively. Both are vectors of length N, andeach input vector is also termed OFOM symbol. F denotesthe OFT matrix with elements:F= _1_e-2j1rmn/Nm,n VN .The frequency domain noise Wk ~ FtWk is a zero meancomplex Gaussian vector with covariance matrix: Cw = I.The multiplication by VN follows from the definition of theFourier analysis matrix, (2).The channel itself is better characterized in the time domainwhere H k= [h k0' ... ,h k £_1,0, ... ,O]T, and L istermed the channel memory length. We assume a wide sensestationary Gaussian channel, with uncorrelated scattering(WSSUS channel). The stacking of the channel vector hasa complex Gaussian distribution, H~ rv CN(E[H~],CHk),and the covariance matrix of the channel (in time domain)can be written as:where C~ is a (k + 1) x (k + 1) Toeplitz matrix defined by(Ckh)l m = C h h jC h , CH is an N x N diagonal, O,lN' OzmN 0,0 0matrix and ® symbolIzes the Kronecker product. For normalizationpurposes we will use Tr[CH ] o= 1. We also define thek x 1 vector c~ in which (C~)l = (C~)l,k, l = 0, ... , k - 1.This vector is the correlation between the channel tap valueat the k-th symbol and its value at all previous symbols. The(2)(3)vector will be used in the analysis of the channel estimationofthe k-th symbol. We also assume that at most one channeltap has an expectation different from zero.The input signal has constraints on the first and secondmoments ofthe symbol power:E[X!X k ] ~ Npx, E[(X!X k )2] ~ aN 2 p;. (4)where a ~ 1 is termed the quadratic constraint constant.In the journal version of this paper we also consider a peakpower constraint and the case in which the transmitted symbolsare required to be statistically independent. In this workwe do not allow any feedback, and therefore we assume thatthe input symbols are statistically independent ofthe channel.3. CHANNEL CAPACITY AND BOUNDSAs the channel capacity is not known, in the following wepresent tight lower and upper <strong>bounds</strong> on it. Due to the spacelimitations, some results are stated without proof.Theorem 1 The capacityofthe channel is upperboundedby:UBI = max E1:s;,a:S;QC ~ min (LB 1 ,LB 2 ) (5)LB I = EfI o[lOg (NPxlho,oI2 + 1)]2LB = N IE [fi ]1ap;(T c - N) aNp;2 px n,O + 1 + Px eTc _ N) + 2where T c is the effective coherence time given in (7).ProofofTheorem 1 is given in the journal version ofthispaper [11}.Note that LB 1 is the complete CSI bound and is dominantin high SNRs, while LB 2 takes into account the uncertaintydue to the missing CSI, and is dominant for lower SNRs.Theorem 2 The capacity ofthe channel is lower boundedby:IN(3E [fi ]+ 1.. pxrrc-N) Zl2n,ON Px(Tc-N)+£UB2 ~ E log 1 + 1 L-Np-x + -N-Px-(""'lC"'Tc---N-)-+-N-Lwhere z rv CN(O, 1) is a standard complex normal randomvariable, and CQ(p) is the achievable mutual informationforQPSKtransmission overAWGNchannel with SNR equal to p,evaluated in [12}.(6)1-4244-2482-5/08/$20.00 ©2008 IEEE 756 IEEEI2008

Proofof Theorem 2 is given in the journal version of thispaper [iil.Note that UBI is achieved using narrowband QPSK transmissionwith duty cycle of j3, and is tight for low SNRs. Onthe other hand, UB 2 is achieved using wideband continuousGaussian modulation, and is tight for high SNRs ifT e » N.The approximation in UB 2 is based on the approximation presentedin (18) and is shown to be accurate even for T e thatspan as little as 10 OFDM symbols. An exact version ofthisbound, which does not depend on the approximation, is givenin the journal version ofthis paper [11].4. THE EFFECTIVE COHERENCE TIMEThe previous Theorems were presented as a function of thefollowing quantity that was termed effective coherence time:'Tc = }~~ NCht (NPx [C~-l - Chcht] +If1 ch + N.(7)In this section we present some results which explain whyT e was termed effective coherence time, and give some insightsinto its physical nature. We begin by considering aconstant amplitude modulation (e.g., QPSK), and afterwardswe present an approximation that holds for other modulations.The estimation error covariance matrix given the pasttransmitted and received symbols is given by:E{Sxn-1} == Pxl. Writing the estimation error covarianceomatrix, (9), in the time domain, and substituting (3) we have:C nCHn - NPxch t 0 CHn. (NPxC~-1 0 CH n+ I) -1 Ch: 0 CH n. (11)This matrix is a diagonal matrix in which for any channel tapsuch that (CH n)l,l > 0 we can write:(CHn)~l + NPxch: t (12)o(Npx (CHJ1,1 [C~-l- Chch t ]+If\hoWe wish to find a relation between the coherence timeand the channel estimation performance. We therefore tum toa special case ofa block fading channel, in which the channelis constant over its coherence time.As a reference we assume that the n-th symbol is the lastsymbol ofa block. The channel estimation for the last symbolofa block is based on the previous TelN - 1 symbols, whichhave exactly the same channel impulse response as the n-thone. In mathematical terms, the channel covariance matrix is:n - TelN < i,j :S notherwise(13)and the inverse of the estimation error of each channel tap,(12), becomes:where Sx is the spectrum of the symbol X given by: Sx ==diag(X)diag(X)t. We define the following covariance matrixwhich is strongly related to the effective coherence time,by replacing the transmitted spectrum with its expectation:(8)We use this model for a reference, because its estimation performanceare directly related to its coherence time.The direct relation between the coherence time and thechannel estimation error variance exhibited by the "block"fading model can be used to define an effective coherencetime for a general channel model. We define the effective tapcoherence time ofan arbitrary channel as:Cil - NCiln-l iI t E {Sxn - 1 }nO' n 0. (NCiln-l E {Sxn-1} + I) -1 CHn~l H0o 0 0 , n4.1. Wideband constant amplitude modulation(9)(c )-1 (C )-1n l,l - H n l,lN + 1· 1mn~oo PxN + N lim ch: tn~oo(15)A constant amplitude modulation can be caharcterized as amodulation in which the transmission spectrum is equal to itsexpectation, Sxn-1 == E{Sxn-1}. Therefore, for a constanto 0amplitude modulation we have:(10)In this subsection we consider the special case of widebandconstant amplitude modulation, characterized bySxn-1 ==owhere the second row in the equation has made use of(12).This definition ofeffective tap coherence time seems useful,since it has a direct relation with the channel estimationperformance. However, it is important to note that this effectivetap coherence time depends on the channel tap power,(C Hn )l,l, and will in general be different for each channel tap.In fact, the effective tap coherence time is a non-increasingfunction ofthe channel tap power. Taking an engineering perspective,one tends to consider the worst case as the preferred1-4244-2482-5/08/$20.00 ©2008 IEEE 757 IEEEI2008

channel measure. Therefore, it is interesting to note that theeffective coherence time, defined in (7) satisfies:(16)with equality in the case that a single channel tap contains allofthe channel energy.Thus, The effective coherence time, which was defined in(7), and appeared in the capacity <strong>bounds</strong>, is a good characterizationofthe channel in terms ofthe achievable estimationperformance. To be precise, it represents the coherence timeof a block fading channel model in which a system that applieswideband constant amplitude modulation achieves thesame estimation error variance.4.2. Narrowband constant amplitude modulationAnother case for which we can provide a close-form expressionfor the estimation performance is the narrowbandconstant amplitude modulation, characterized by Sxn-1 ==oE{Sxn-1} == I ~ So, where So == diag([I, 0, ... ,0]). In thisocase we get:lim (en)n---+oo l,l4.3. Wideband approximation1 1N 1 +PxCTe- N)'(17)Thus, the effective coherence time also directly characterizethe estimation performance in the narrowband case.For other modulations, we use we use the following approximationfor the estimation error covariance matrix which isaccurate if the input symbols are iid and the effective coherencetime is long enough:(18)This approximation is especially convenient as it dependsonly on the input statistics and not on the actual transmittedsymbols, as is for (8).As long as this approximation holds, the physical interpretationsof the effective coherence time presented in subsections4.2 and 4.1 holds for any modulations type. Furthermore,this approximation was used to prvide the simple formofUB 2 which is important in the high SNR regime.4.4. Properties ofthe effective coherence timeIn contrast with the channel coherence time, the effective coherencetime is also a function of the system SNR (a nonincreasingfunction ofthe SNR). This is negligible in the lowSNR extreme, where the effective coherence time reaches itsmaximal value. At the limit when the SNR goes to zero all the_y=0.866(TcO=120)- • - y=0.949(TcO=300) .,. -8-· y=0.985(TcO=990)103~::.~:-:~:~~:~.:~~. ~~5.::~ ,,: ;."..,.::: ~:.: '.' '.':.:'.': :~: ···.0 .. · ..0 .. y=0.995(TcO=3000) :::.. · ~·s ·····0: .102~ :.....:.. :-:~ ..:. ······..0:··........ ~ i········ .: : ~ Ii. .. :. ··..·e···:...~. ·f·~· '1'~' *.~~. ~~.~ .~.~ ~,'." ....:~ :~....:.......""'... 151.",~ ~. ",,~, : ''EI." : '0.. :: :: :- : -:....:..:....."."'.~.~~5.-2.,~!~~10 1 L.---..L-_.....I...---JL.-----L..._....L.---I._---L....._~_____L_--L.------l-50 -45 -40 -35 -30 -25 -20 -15 -10 -5SNR[dB]Fig. 1. Effective coherence time for the ARI channel modelfor various parameter values.effective tap coherence times are equal and reach their upperbound of:Teo == lim Te == N lim (Ch t ch + 1). (19)Px---+On---+ooThis value is equivalent to the channel coherence time definedin [8]For higher SNRs in most channel models the effective coherencetime also depends on the SNR, and decreases as theSNR increases. At first this may seem as a surprising characteristicof a coherence time. intuitively one would expectthe coherence time to characterize the channel, regardless ofthe system parameters. The dependence on the system SNRarises from the fact that the effective coherence time actuallycharacterize the estimation performance.Intuitively one can say that a system with higher SNRwould require better channel estimation. Thus, such a systemwill be able to use only channel measurements that has highercorrelation to the present channel state. Further increase in theSNR will further reduce the number ofuseful measurementsand therefore reduce the effective coherence time.Let us consider another simple channel model, termedAR 1 (auto regressive channel model of the first order). Thischannel model is defined by a single parameter, "(, the channelforgetting factor, and is characterized by C H .,H. ==t J"(Ii-jlC Ho . The effective coherence time for this channelcan be calculated using (7), and its low SNR limit isTeo == N / (1 - ,,(2).5. NUMERICAL EVALUATIONIn this section we present numerical results that demonstratethe behavior ofthe effective coherence time and the tightnessof the capacity <strong>bounds</strong>. For these calculations, we assumeE[H n ] == 0, equal power taps, i.e., (CHN)l,l == I/L for1-4244-2482-5/08/$20.00 ©20081EEE 758 IEEEI2008

7. REFERENCES[1] G. Taricco and M. Elia, "Capacity of Fading Channelswith no Side Information," IEEE Electronic letters,vol. 33, no. 16, pp. 1368-1370, July 1997.[2] I. Abou-Faycal, M. Trott, and S. Shamai, "The CapacityofDiscrete time Memoryless Rayleigh Fading Channels,"IEEE Trans. on Information Theory, vol. 47, no. 4,pp. 1290-1301, May 2001.~ UB, Teo-300010- 6w", ../.-h/"' ,', , " j - e - LB, Teo=3000__ UB, T eO=300F·.· ,· · · ~ - tf- -LB, T eo=30010- 7 L----L.-_...L-----l...._~___L_-L-------l._..J===c::::::=r::::====.J-50 -45 -40 -35 -30 -25 -20 -15 -10 -5SNR [dB]Fig. 2. Capacity <strong>bounds</strong> vs. SNR for ARI fading channel(Teo == 3 . 10 2 , 3· 10 3 ) and quadratic power constraint.o ~ 1 < L. We set the OFDM symbol length to N == 30samples, and the channel memory length to L == 5 samples.The quadratic constraint constant in (4) is set to a == 10.Figure 1 depicts the effective coherence time (T e ) of theARI channel model as a function ofSNR (Px) for various valuesof its parameter (1). As can be seen, the effctive coherencetime is constant for low enough SNR and then descendsas SNR increases. Note that the smallest effective coherencetime in this model is equal to N (meaning that the differentOFDM symbols are practically independent). Our <strong>bounds</strong> areusefull as long as the effective coherence time is much largerthen the symbol length (T e » N)Figure 2 depicts the <strong>bounds</strong> on the channel capacity oftheARI channel when the value of the channel parameter is setto 1 == 0.949 (Teo == 300) and 1 == 0.995 (Teo == 3,000). Ascan be seen, the <strong>bounds</strong> are tight for all SNR values. For verylow SNR both <strong>bounds</strong> approximately equal ap;Te. For highSNR the <strong>bounds</strong> are tight as long as T e » N.6. SUMMARYIn this paper we analyzed the capacity ofOFDM under spreadchannels, and presented tight <strong>bounds</strong> on the channel capacity.These <strong>bounds</strong> are characterized by the newly defined effectivecoherence time parameter. The effective coherence timecharacterize the number of effective number of samples thatcan be used for channel estimation. The presented <strong>bounds</strong>are tight for any SNR. For low SNR the <strong>bounds</strong> converge toap~Te, while for high SNR the complete CSI ('known channel')upper bound is tight as long as the effective coherencetime is large enough.More details on the <strong>bounds</strong> derivation are given in [11].[3] S. Verdu, "On Channel Capacity per Unit Cost," IEEETrans. on Information Theory, vol. 36, no. 5, pp. 1019­1030, September 1990.[4] A. Lapidoth and S. Shamai, "Fading Channels: HowPerfect Need "Perfect Side Information" Be?" IEEETrans. on Information Theory, vol. 48, no. 5, pp. 1118­1134, May 2002.[5] A. Lapidoth and S. M. Moser, "Capacity Bounds ViaDuality With Applications to Multiple-Antenna Systemson Flat-Fading Channels," IEEE Trans. on InformationTheory, 2003.[6] A. Lapidoth, "On the Asymptotic Capacity of StationaryGaussian Fading Channels," IEEE Trans. on InformationTheory, vol. 51, no. 2, pp. 437-446, February2005.[7] I. E. Telatar and D. N. C. Tse, "Capacity and MutualInformation of Wideband Multipath Fading Channels,"IEEE Trans. on Information Theory, vol. 46, no. 4, pp.1384-1400, July 2000.[8] M. Medard and R. G. Gallager, "Bandwith Scaling forFading Multipath Channels," IEEE Trans. on InformationTheory, vol. 48, no. 4, pp. 840-852, April 2002.[9] 1. G. Proakis, Digital Communications. McGraw-Hill,1995.[10] D. Tse and P. Viswanath, Fundamentals of WirelessCommunication. Cambridge University Press, 2005.[11] I. Bergel and S. Benedetto, "Bounds on the Capacity ofOFDM Under Spread Channels," Submittedforpublicationin the IEEE Trans. on Information Theory.[12] R. S. Kennedy, Fading Dispersive CommunicationChannels. Wiley-Interscience, New York, 1969.1-4244-2482-5/08/$20.00 ©2008 IEEE759IEEEI2008